Train Shunting on a Tree

The problem is set in a railroad shunting yard where a train (a sequence of connected carriages) is represented as a simple path in a tree. The tree T has n vertices with edges connecting them. A train is a simple path (no repeated vertices or edges) of fixed length m (i.e. it contains m edges) along the tree.

The train moves in phases. In one phase, you are allowed to move one end of the train along an edge adjacent to that end provided that the new vertex is not already on the train. Simultaneously, the carriage at the opposite end is removed so that the train always remains of length m. More precisely, if the train is represented by the vertex sequence $$v_0, v_1, \ldots, v_m,$$ then in one move you may either:

• Choose the left end (v₀) and a neighbor \( w \) of \( v_0 \) such that \( w \neq v_1 \) and \( w \notin \{v_0,v_1,\ldots,v_m\}, \) then form the new train: $$w, v_0, v_1, \ldots, v_{m-1}.$$
• Or choose the right end (v_m) and a neighbor \( w \) of \( v_m \) such that \( w \neq v_{m-1} \) and \( w \notin \{v_0,v_1,\ldots,v_m\}, \) then form the new train: $$v_1, v_2, \ldots, v_m, w.$$

You are given the tree T with n vertices (numbered from 1 to n) and all n-1 edges; you are also given two paths P and Q of the same length m (specified by their two endpoints – note that the unique simple path between the two endpoints in a tree gives you the whole train). It is guaranteed that paths P and Q do not share any edge. Your task is to determine whether it is possible to transform P into Q using a sequence of moves, and if so, find the minimum number of moves required. A configuration is considered matching Q if the train occupies exactly the same edges as the unique simple path corresponding to the given endpoints of Q (note that the train may be in either orientation).

You need to implement the following two functions without performing any input/output operations:

void init(int n, vector X, vector Y);

This function is called once at the beginning. The tree has n vertices, and the two vectors X and Y (each of size n-1) define the edges of the tree as \((X[i], Y[i])\).

long long train(vector Z);

The parameter Z is a vector of size 4 representing the endpoints: Z[0] and Z[1] are the endpoints of the initial train position P, while Z[2] and Z[3] are the endpoints of the target train position Q. The function should return the minimum number of moves required to transform P into Q. If the transformation is impossible, return -1.

Note: In the tree, the unique simple path between any two vertices defines the positions of the train. The train may be moved from either end as long as the move does not result in a repeated vertex in the train.

inputFormat

There is no standard input. Instead, the system will call the two functions init and train directly. First, init(n, X, Y) is called once to initialize the tree with n vertices and n-1 edges given by vectors X and Y. Then, one or more calls to train(Z) will be made. Each call to train provides a vector Z of four integers: the endpoints of the initial path P and the target path Q.

outputFormat

For each call to train, return a single integer: the minimum number of moves to transform the initial train configuration into the target configuration. If it is impossible, return -1.

sample

n = 5
X = [1, 2, 3, 4]
Y = [2, 3, 4, 5]
Call: train([1, 3, 3, 5])

2